Question

Suppose you are a fund manager managing a portfolio worth $10million with Beta equal 1.2. The index futures price is 1000 and each future contracts is on $50 times the index. If you want to keep the value of the portfolio stable without selling the portfolio in the next two months, what is your hedging strategy? In the maturity date, the index is 1050, please show the success of your strategy. The risk-free interest rate is 5% per annum (continuously compounded).

Answer 1

First let us understand, what is hedging and why we are doing it?

Hedging is simlpy saying to hedge your naked postion, which in this case is your long portfolio of $10million.

Let us say, you are expecting a fall in the market, but if you were to sell all of your portfolio, it will involve huge amount of transaction costs and brokerage.

So, instead you can short-sell the index futures (Remember! taking the opposite positon vs. your portfolio).

Calculting the hedging value

Now, note that the hedging value would not be directly your portfolio value. Since, the portfolio beta is 1.2, it means if market falls by 5%, your portfolio falls by 5*1.2=6%.

So we have to short sell $10million X 1.2 = $12million of index futures.

No. of contracts required

Now to short $12million of futures, we need to calculate the no. of contracts required.

One contract is $50 X Index price, i.e. $50 X 1000 = $50,000.

No. of contracts required to short = $12million / $50,000 = 240

Please note, we would receive 240 contracts X $50 = $12,000.

So, the answer of first part is, our hedging strategy would involve selling 240 contracts of index furtures in order to protect our portfolio from loss.

Now we have to see the success of our strategy

After two months, Index is at 1050, so it gained 50/1000 = 5%.

Since our portfolio beta is 1.2, it must have gained 5% X 1.2 = 6%.

So, our portfolio value after two months would be $10million X 1.06 = $10.6million, a gain of $0.6million.

But on the other hand, we have lost on our future short. At maturity, we need to close our position by buying 240 contracts at 1050, i.e. we would loose 50 (gain in Index) X 240 contracts X $50 (intial value) = $0.6million.

So gain of the portfolio is nullified by our postion on index futures.

But there is one last part, by shorting index, we received $12,000. We would have earned 5% per annum interest on that.

So, after two months, we will receive $12000 X, where a = 5% and x = time in years, i.e. $12,100.

You would notice we still have gained $100.